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Next: MC450 Finite Element Methods for Partial Up: Year 4 Previous: MC440 Commutative Algebra

MC446 Algebraic Topology

MC446 Algebraic Topology

Credits: 20 Convenor: Dr. J. Hunton Semester: 1

Prerequisites: essential: MC145, MC240, MC241 desirable: MC242 or MC249 or MC254 or MC255
Assessment: Coursework: 10% Three hour examination in January: 90%

Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

This module will draw on some basic ideas from both algebra and analysis. In algebra knowledge will be assumed of polynomials and of the number systems the integers, the rationals and the integers modulo n; these concepts are all introduced in MC145; familiarity with the basic concepts of vector spaces, as covered in MC241 will also be required. The algebraic concepts discussed in MC242(MC255) or MC249(MC254) may prove helpful, especially the ideas of quotient structure. In analysis the central topics drawn on are the ideas of topological spaces, of closed or compact subsets of ${\bf R}^n$ and continuous functions. These are all covered in MC240.

Course Description

Fact: if you have a dog which is completely covered in hair, then there is no way of combing that hair smooth so that there is no parting or bald spot. This is the so-called `hairy dog theorem'.

Fact: no matter how badly you make a sandwich out of two pieces of bread and a slice of ham then it is always possible to find a plane cutting the sandwich which bisects exactly each piece of bread and the slice of meat. This is the so-called `ham sandwich theorem'.

Fact: if you associate to each point of the Earth's surface the two numbers (t,p) where t is the temperature at that point and p is the air pressure there, then there is always at least one point which has the same values of t and p as its diametrically opposite point. This is the `Borsuk-Ulam theorem'.

Fact: reef knots and granny knots are distinct types of knots and no matter how you fiddle with one, short of untying it and retying it as the other, you can't transform the first into the second.

These are all very deep geometric facts about the shapes of dogs, ham sandwiches, the Earth and knots; they are especially deep as they are true for any shape or size of dog, sandwich, planet etc., and they are all proved by a remarkably successful set of mathematical ideas and methods that date from early this century. These ideas, and ones like them, constitute the subject of Algebraic Topology.

The basic idea of the subject is to find a formal way of translating geometric problems into an appropriate algebraic language. If this is done successfully then the geometric problem is usually reduced to a fairly simple piece of algebra and the problem is solved or the theorem proved by relatively trivial algebra. In this module we shall prove each of the facts above by translating the underlying geometry into questions about integers, polynomials or vector spaces.

Aims

This module aims to introduce the basic ideas of algebraic topology and to demonstrate its power by proving some memorable theorems.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To be able to solve some routine problems in low and high dimensional topology.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

The ability to apply taught principles and concepts to new situations.

Syllabus

Knot theory: knots, equivalences of knots, Reidermeister moves. Examples of knots. The Jones polynomial and Kauffman's proof of its invariance. Properties of the Jones polynomial and its computation. Distinction of knots by application of the Jones polynomial.

Homology theory: elementary categorical concepts - category, functor. Topological spaces, homotopy equivalent spaces, polyhedra and the simplicial approximation theorem. Examples of standard spaces. Simplical and singular homology groups, their functorial properties and computation (the Eilenberg-Steenrod axioms). Application of homology theory to Brouwer's fixed point theorem, the Ham Sandwich theorem, the hairy dog theorem and the Borsuk Ulam theorem. Other theorems as time allows.

Reading list

Recommended:

M. A. Armstrong, Basic Topology, Springer. W. Lickorish, An introduction to Knot Theory, Springer.

Background:

W. S. Massey, A Basic Course in Algebraic Topology, Springer. H. Sato, Algebraic Topology: an intuitive approach, Springer.

Details of Assessment

There will be a series of pieces of work set during the semester which together will count for 10% of the final mark.

There will be six questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to four questions. All questions will carry equal weight.

Next: MC450 Finite Element Methods for Partial Up: Year 4 Previous: MC440 Commutative Algebra

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Last updated: 10/4/2000
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